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MA519_JAN98 - 4 A point P is picked at random(i.e uniformly...

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Math 519 Qualifying Exam-January 10, 1998 Each part of each problem is worth 17 points. A table of the standard normal distribution is attached. 1. Ten points are selected independently and at random(i.e. according to a uniform (0,1) distribution) from the interval (0,1). Let D be the minimum of the ten distances from these points to the complement of (0,1). Find the expectation of D . 2. Six cards are dealt from a shuffled deck, one at a time. (a) Find the probability of “two triples,” that is, only two of the demominations ace,2,...J,Q,K, are present in the hand, and exactly three of each of these denom- inations occurs. (b) Find the expected number of suits among the six cards. (The suits are hearts, clubs, spades, diamonds.) 3. A balanced six sided die numbered 1-6 is rolled 1000 times. Estimate the probability that the sum of all the even numbers rolled exceeds the sum of all the odd numbers rolled by at least 480.
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Unformatted text preview: 4. A point P is picked at random (i.e. uniformly with respect to area) on the surface of the unit sphere x 2 + y 2 + z 2 = 1 of Euclidean three space. Then a point Q is picked at random (uniformly with respect to length) on the line which connects the origin and P. (a) Let ( X,Y,Z ) be the rectangular coordinates of Q . Find the joint density of ( X,Y,Z ). (b) Show that the random variables X,Y,Z of part (a) are not independent. Suppose P is picked as above, and that a point R is picked so that it is on the half line starting at the origin and going through P on out to infinity, with the distance of R from the origin being a random variable with a density function f . (So, the special case where f equals one between 0 and 1 gives the point in part (a).) Let ( X ,Y ,Z ) be the rectangular coordinates of R . Is there any choice of f which makes X ,Y ,Z independent? 1...
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